A new method of solving variational problems in the theory of quasiconformal mappings (Q1114809)

An analytic function \(w=F(\zeta_ 1,...,\zeta_ n)\) of n complex variables is given. It is assumed that n distinct points \(z_ 1,z_ 2,...,z_ n\) in \({\mathbb{C}}-\{0,1\}\) are selected and that \(F(z_ 1,...,z_ n)=0\). Let Q be the space of quasiconformal homeomorphisms of the sphere \({\bar {\mathbb{C}}}\) which fix the points 0, 1, and \(\infty\) and Q(k) the subspace of Q consisting of mappings which are \((1+k)/(1-k)\)- quasiconformal. For f in Q, we denote by F(f) the value \(F(f(z_ 1),...,f(z_ n))\). In general, the author considers the problem of finding the supremum of \(| F(f)|\) taken over f in Q(k). Let \[ c_ j=F_{z_ j}(z_ 1,...,z_ n)\quad and\quad \phi_ 0(z)=\frac{1}{z(z-1)}\sum^{n}_{1}\frac{c_ jz_ j(z_ j-1)}{z-z_ j}. \] Theorem. Assume \(\phi_ 0(z)\) has only even order zeroes. Then there exists a constant \(k(F)>0\) depending on F such that for all \(k\leq k(F)\), one has the following inequality: \[ \max_{f\quad in\quad Q(k)}| F(f)| \leq \max_{| t| =k}| F(f^{t\mu})| \equiv d(k), \] where \(\mu =| \phi_ 0(z)| \phi_ 0(z)\) and \(f^{t\mu}\) is a normalized quasiconformal mapping with Beltrami coefficient \(t\mu\). The method of proof is to use the equality of the Kobayashi-Teichmüller metric with the Carathéodory metric on Teichmüller disks of the special form described in the theorem. The conclusion of the theorem holds true on any Teichmüller disk where this equality is in force.